Lagrange's theorem gives a strong test for possible subgroup orders. Once the order of a finite group is known, the order of every subgroup must be one of its positive divisors. This lecture focuses on that restriction. We will not claim that every divisor gives a subgroup; instead, we learn how Lagrange's theorem rules out impossible subgroup orders. This is a basic but essential skill in finite group theory.

Let $(G,\circ)$ be a finite group. A positive integer $d$ is called a $\textbf{possible subgroup order by Lagrange's theorem}$ if $$\begin{equation*} d\mid |G|. \end{equation*}$$ This means that Lagrange's theorem does not rule out a subgroup of order $d$.